No ordinary polynomial problem

Algebra Level 5

The fifth degree monic polynomial $$f(x)$$ satisfies the following conditions.

$$$\begin{split} f(1) & = 721 \\ f(2) & = 1201 \\ f(3) & = 1441 \\ f(4) & = 1261 \\ f(5) & = 673 \end{split}$$$

What is $$f(6)$$?

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